Spin chains with symmetry-protected edge zero modes can be seen as
prototypical systems for exploring topological signatures in quantum systems.
These are useful for robustly encoding quantum information. However in an
experimental realization of such a system, spurious interactions may cause the
edge zero modes to delocalize. To stabilize against this influence beyond
simply increasing the bulk gap, it has been proposed to harness suitable
notions of disorder. Equipped with numerical tools for constructing locally
conserved operators that we introduce, we comprehensively explore the interplay
of local interactions and disorder on localized edge modes in the XZX cluster
Hamiltonian. This puts us in a position to challenge the narrative that
disorder necessarily stabilizes topological order. Contrary to heuristic
reasoning, we find that disorder has no effect on the edge modes in the
Anderson localized regime. Moreover, disorder helps localize only a subset of
edge modes in the many-body interacting regime. We identify one edge mode
operator that behaves as if subjected to a non-interacting perturbation, i.e.,
shows no disorder dependence. This implies that in finite systems, edge mode
operators effectively delocalize at distinct interaction strengths. In essence,
our findings suggest that the ability to identify and control the best
localized edge mode trumps any gains from introducing disorder.
Anonymous Report 3 on 2019-3-15
"1. Referring to both, the prefactor J in front of the perturbation and the parameter η, as an interaction is a bit unfortunate and might be confusing: For instance, in the caption to
Fig. 2c) (last sentence in figure caption) the authors talk about ”non-interacting results”
(meaning η = 0) presented as a function of interaction strength J in Panel c."
In order to clarify this confusion, we extended the discussion of the Hamiltonian act-
ing as a perturbation introduced in Eq. (4). Furthermore, we added an additional dis-
claimer in the caption that the interaction strength may here also be referred to as
"2. I did not understand how the time-dependent quantity B_0(t) enters, see Eq. (15), and what is its meaning. Also replacing the time average there by a basis expansion looks like invoking an ergodicity argument, e.g. equilibration. But is this justified within this study of localization effects?"
The operator given in Eq. (15) is the equilibrium representation of any B for non-
degenerate systems. Due to the topological degeneracies, we also require a rediago-
nalization of the subspaces in the basis of the constants of motion as explained below
Eq. (15). While this is basically only a definition, it is not a priori clear, that an operator
will equilibrate towards this form under time evolution. This is however expected for
interacting systems, even such that localize as we now also discuss below Eq. (15) and
cite the corresponding result. These operators serve as an ansatz for our algorithm
only and even work in a regime, where equilibration is not expected (non-interacting,
"3. In Fig. 1c the color coding is nearly not distinguishable in the symbols forming the
We have added a comment on this in the caption.
"4. Fig. 3c): Color code given in upper right box does not coincide with the red and blue
symbols used in the panel."
Here, the red symbols are actually on top of the orange ones, which we now also
indicate in the caption.
"5. Fig. 2a,b): Where does the symmetry with respect to site number comes from that does not exist for η = 0, see Fig. 1?"
We thank the referee for pointing this out. The symmetry is inherited from the sym-
metry of the set S we use to calculate the support. Since for the interacting system,
we always need to construct products of edge modes, the support is predominantly
on both edges. This is why we cut out boxes centered around the middle of the chain
to calculate the support which in turn causes the symmetry in our plots. For compar-
ison: In the non-interacting case, we work with the probably more intuitive shape of
S which is just a consecutive region starting at one edge. We have added an explanation towards the end of section 3.3 and a discussion in the results section (4.2) of the interacting model. Further comments on typos were fixed.
Anonymous Report 2 on 2019-3-3
"Unfortunately, the extensive use of mathematical notation and jargon makes it quite
hard to read and I strongly suggest to explain the main ideas and the method in plain
english before casting them into equations. Also, the notation should be explained this
way, examples are the set complements, first featuring in Eq (16), as well as what stands
behind Eq. (14), which is unclear to me. Why is there a floor function (?) appearing here?"
We thank the referee for this feedback and overhauled the presentation of mathemat-
ical details according to it. As this was also in similar fashion pointed out by the other
referee, we added an explanation of the sets used for the support calculation in section
3.3. Moreover, we put a more colloquial explanation below Eq. (14).
"Claims about exponential (in what?) corrections of the vanishing commutator of the
edge mode operators with the Hamiltonian were made, and I would find it useful to see
a numerical verification of this claim."
Indeed, we missed to write ”exponential in system size” at one point in the document
and are thankful for the referee to spot this error. The failure of the edge modes to
commute with the Hamiltonian is in fact guaranteed by Kramer’s theorem. A clarification as to why this is the case has been added to Section 2. Investigating this numerically in the η = 1 case is not possible, as we would need to recover the individual edge modes, and not the bilinears which are available to us.
"The description of the algorithm for the construction of the edge modes is really not clear and I think it should be extended considerably and explained in more detail, not relying on the previous work by the authors (one particularly unclear point is why the order of eigenvectors should matter)."
In order to clarify this part, we have moved all reference to the MBL construction
towards the end of section 3.2 and added more detail to the main steps of the algorithm.
Specifically, we now point out that since any constant of motion can be mapped onto
the Pauli-Z operators only the order of the eigenvectors determine its locality.
"The decay of the support with decreasing subsystem size is used to quantify the decreasing operator weight at long distances. It would be useful to add an illustration of the subsystems used here."
This points is similar to the first one, which we hope to sufficiently incorporate by
putting in additional description of the sets.
"The support does not decrease smoothly, there are considerable jumps and plateaus visible. Is it clear where they stem from?"
While we previously ascribed these plateaus intuitively to the three-locality of the
Hamiltonian, this remark motivated us to actually show its origin for the non-interacting perturbation in the infinite systems limit. The calculation can be found in an additional appendix and we furthermore added explanatory sentences into the main text.
"I think what is dearly missing in this analysis is a careful study of the dependence of the results on system size. It is clear that there is considerable leakage of certain edge modes in the bulk of the system and it is therefore crucial to increase the size as much as possible, given the state of the art of full diagonalization, I believe that at least system sizes up to L=15 should be feasible for this 2^L Hilbert space. This would add at least another data point for the analysis of the exponential decay and make it more plausible. In addition, the stability with system size should be analyzed, comparing results for different chain lengths."
We agree with the referee that in the previous version, we missed out on providing
standard evidence for the reliability of our algorithm. In the present resubmission
we added system size scaling for the non-interacting code. Due to plateau structure
that also persists in the interacting case we essentially only obtain very few points
for fitting the exponential envelope. Making the system size smaller will reduce this
to only two points, which results in very unstable data. Enlarging system size on the
other hand is not feasible either because the bottle neck is unfortunately not the exact
diagonalization but rather the rediagonalization of the 2^(L−2) many 4 × 4 matrices. A
single realization takes 8 days to construct all three edge modes on 12 sites. We have
added an explanation of this problem to the resubmission as well.
"In Fig. 1 right there appears to be an artifact in the error bars (?), which vanish suddenly at 10^−4 . What is the reason for this?"
We are thankful for this remark as we missed putting an explanation into the first
submission. The error bars are caused by a very sharp dropoff of the support due to
the small interactions which lead to a less stable fit. We added such an explanation in
the result description in section 3.1 as well.